Finite Group Databasebeta

← #45664418

Subgroups — #45664418

Order 32896.

306 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#117×D257C211yesmaximal, characteristic
2#30656700C211yesmaximal, characteristic
3C4×#16861595C211yesmaximal, characteristic
4#30656498C211yesmaximal, characteristic, complemented
5#30656497C211yesmaximal, characteristic, complemented
6#30656496C211yesmaximal, characteristic, complemented
7#339×C257C211yesmaximal, characteristic, complemented
8C22×C4×D257C2211yescharacteristic
9C2×#16861602C2211yescharacteristic
10#22551551C2211yescharacteristic
11#22551550C411yescharacteristic, complemented
12#22551550C411yescharacteristic, complemented
13#22551549—21no—
14#22551548—21no—
15C4⋊C4×D257C411yescharacteristic, complemented
16C4⋊C4×D257C411yescharacteristic, complemented
17C2×#16861595—21no—
18C22⋊C4×D257—21no—
19C2×C4×Dic257C2211yescharacteristic
20C42×D257—21no—
21#22551504—21no—
22#22551504—21no—
23#22551499—21no—
24C2×M16×C257C2211yescharacteristic
25#117×C257C2211yescharacteristic
26C2×C42×C257C2211yescharacteristic
27C23×D257D411yescharacteristic
28C22×Dic257—21no—
29C22×Dic257D411yescharacteristic
30C2×C4×D257—21no—
31C2×C4×D257—21no—
32C2×C4×D257—21no—
33C2×C4×D257—21no—
34C2×C4×D257—21no—
35C2×C4×D257—21no—
36C2×C4×D257C2×C411yescharacteristic
37C2×C4×D257C2×C411yescharacteristic
38C2×C4×D257D411yescharacteristic
39C2×C4×D257D411yescharacteristic
40#16861602—21no—
41#16861602—21no—
42C2×#14523339—21no—
43C4⋊Dic257C2×C411yescharacteristic
44C4⋊Dic257C2×C411yescharacteristic
45#16861597—21no—
46#16861597—21no—
47C22⋊Dic257—21no—
48#16861595—24no—
49#16861595—21no—
50#16861595—21no—
51C4×Dic257—21no—
52C4×Dic257—21no—
53C4×Dic257—21no—
54C4×Dic257—21no—
55C4×Dic257—21no—
56C22×C4×C257—21no—
57C22×C4×C257C2311yescharacteristic
58M16×C257—21no—
59M16×C257—21no—
60C2×C8×C257—21no—
61C4⋊C4×C257C2×C411yescharacteristic
62C4⋊C4×C257C2×C411yescharacteristic
63C22⋊C4×C257—21no—
64C42×C257—21no—
65C42×C257—21no—
66C42×C257—21no—
67C22×D257—21no—
68C22×D257—21no—
69C22×D257—22no—
70C22×D257—41no—
71C22×D257C22⋊C411yescharacteristic
72C22×D257C22⋊C411yescharacteristic
73C2×Dic257—21no—
74C2×Dic257—21no—
75C2×Dic257—22no—
76C2×Dic257—21no—
77C2×Dic257—21no—
78C2×Dic257—41no—
79C2×Dic257—41no—
80C2×Dic257C22⋊C411yescharacteristic
81C2×Dic257C22⋊C411yescharacteristic
82C4×D257—21no—
83C4×D257—21no—
84C4×D257—21no—
85C4×D257—21no—
86C4×D257—21no—
87C4×D257—21no—
88C4×D257—21no—
89C4×D257—21no—
90C4×D257—42no—
91C4×D257—42no—
92#14523339—41no—
93C23×C257C2×D411yescharacteristic
94C2×C4×C257—22no—
95C2×C4×C257—21no—
96C2×C4×C257—21no—
97C2×C4×C257—41no—
98C2×C4×C257—41no—
99C2×C4×C257C22×C411yescharacteristic
100C2×C4×C257C2×D411yescharacteristic
101C2×C4×C257#4111yescharacteristic
102C2×C4×C257#4111yescharacteristic
103C2×C4×C257#4111yescharacteristic
104C2×C4×C257#4111yescharacteristic
105C8×C257—41no—
106C22×C257—21no—
107C22×C257—21no—
108C22×C257C2×C22⋊C411yescharacteristic
109C22×C257#12311yescharacteristic
110C22×C257#12311yescharacteristic
111C2×D257—21no—
112C2×D257—21no—
113C2×D257—21no—
114C2×D257—21no—
115C2×D257—21no—
116C2×D257—42no—
117C2×D257—42no—
118C2×D257#10412yes—
119C4×C257—42no—
120C4×C257—41no—
121C4×C257—41no—
122C4×C257#11711yescharacteristic
123C4×C257#11711yescharacteristic
124C4×C257#12311yescharacteristic
125C4×C257#12311yescharacteristic
126Dic257—21no—
127Dic257—21no—
128Dic257—42no—
129Dic257—41no—
130Dic257—41no—
131C2×C257—21no—
132C2×C257—21no—
133C2×C257#38111yescharacteristic
134C2×C257C2×#10411yescharacteristic
135C2×C257#42111yescharacteristic
136D257—22no—
137D257—41no—
138C257#182611yescharacteristic, complemented, Sylow
139#1826—2571nomaximal, Sylow
140C2×#117—2571no—
141#407—2571no—
142C4×C22⋊C4—2571no—
143#339—2571no—
144#339—2571no—
145#339—2571no—
146#339D25711yescharacteristic, complemented
147C23×C4—2571no—
148C2×M16—2571no—
149C2×M16C2×D25711yescharacteristic
150#117—2571no—
151#117—2571no—
152#117—2571no—
153#117—5141no—
154#117—5141no—
155#117C2×D25711yescharacteristic
156C2×C4⋊C4—2571no—
157C2×C4⋊C4—2571no—
158C2×C22⋊C4—5141no—
159C2×C22⋊C4—5141no—
160C2×C42—2571no—
161C2×C42—5141no—
162C2×C42C2×D25711yescharacteristic
163C22⋊C8—5141no—
164C22⋊C8—5141no—
165#95—5141no—
166C24—2571no—
167C22×C4—2571no—
168C22×C4—2571no—
169C22×C4—2571no—
170C22×C4—2571no—
171C22×C4—2571no—
172C22×C4—21no—
173C22×C4—5141no—
174C22×C4—5141no—
175C22×C4—5141no—
176C22×C4—5141no—
177C22×C4—5141no—
178C22×C4—5141no—
179C22×C4—5141no—
180C22×C4C22×D25711yescharacteristic
181M16—21no—
182M16—21no—
183M16—5141no—
184M16—5141no—
185C2×C8—21no—
186C2×C8—5141no—
187C4⋊C4—2571no—
188C4⋊C4—2571no—
189C4⋊C4—5141no—
190C4⋊C4—5141no—
191C4⋊C4C4×D25711yescharacteristic
192C4⋊C4C4×D25711yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—5141no—
195C22⋊C4—5141no—
196C22⋊C4—5144no—
197C22⋊C4—5141no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—5141no—
202C42—5141no—
203C42—5141no—
204C42—5141no—
205C42—5141no—
206C23—2571no—
207C23—2571no—
208C23—5142no—
209C23—5141no—
210C23—5141no—
211C23—10281no—
212C23D4×D25711yescharacteristic
213C2×C4—2571no—
214C2×C4—2571no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—5141no—
219C2×C4—5141no—
220C2×C4—5141no—
221C2×C4—5141no—
222C2×C4—5141no—
223C2×C4—5141no—
224C2×C4—5141no—
225C2×C4—5141no—
226C2×C4—5141no—
227C2×C4—5141no—
228C2×C4—5142no—
229C2×C4—5141no—
230C2×C4—5141no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—10282no—
234C2×C4—10282no—
235C2×C4—10281no—
236C2×C4—10281no—
237C2×C4C2×C4×D25711yescharacteristic
238C2×C4D4×D25711yescharacteristic
239C2×C4#1686161711yescharacteristic
240C2×C4#1686161811yescharacteristic
241C2×C4#1686161811yescharacteristic
242C2×C4#1686161911yescharacteristic
243C8—41no—
244C8—10281no—
245C22—2572no—
246C22—21no—
247C22—21no—
248C22—5141no—
249C22—5141no—
250C22—5141no—
251C22—5141no—
252C22—5141no—
253C22—10282no—
254C22—10282no—
255C22C22⋊C4×D25711yescharacteristic
256C22#2255157411yescharacteristic
257C22#2255157711yescharacteristic
258C4—5141no—
259C4—5141no—
260C4—42no—
261C4—41no—
262C4—41no—
263C4—10282no—
264C4—10281no—
265C4—10281no—
266C4#2255154911yescharacteristic
267C4#2255155011yescharacteristic
268C4#2255157411yescharacteristic
269C4#2255157711yescharacteristic
270C2—21no—
271C2—21no—
272C2—5142no—
273C2—10281no—
274C2#3065659911yescharacteristic
275C2#104×D25711yescharacteristic
276C2#3065675211yescharacteristic
subgroupG/H
centerC2×C4D4×D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting#339×C257C2
socleC22×C257C2×C22⋊C4
prime radicalC23×D257D4
cosocleC22×C4×C257C23
hypercenter#339D257
hypocenterC257#1826
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #30656498C2semidirect
5. #30656497C2semidirect
6. #30656496C2semidirect
7. #339×C257C2semidirect
11. #22551550C4semidirect
12. #22551550C4semidirect
15. C4⋊C4×D257C4semidirect
16. C4⋊C4×D257C4semidirect
138. C257#1826semidirect
146. #339D257semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117×D2572
#306567002
C4×#168615952
#306564982
#306564972
#306564962
#339×C2572
#1826257